Stability Theory for Hamiltonian Systems

27 Accordingly it can be seen that the crucial terms are the oo-diagonal entries in the Hamiltonian (9.6), or equivalently the terms in the linearized Lagrangian dynamics (9.5) involving the velocities _ u with skew-symmetric coeecients. Since the time of Kelvin such terms have been called gyroscopic, because they are often associated with eeects of rotation, and, in particular, the stability of steady spins of gyroscopes. The lengthy discussions in 23], Article 345, and 5], Chapter V, consider general perturbations of equations of the form (9.5) with ^ L _q _q positive deenite. See also 24], page 333, where derivative formulas are given for eigen-values arising from modes of (9.5) when subject to general perturbations, but only for the case of semisimple eigenvalues. The inertia theorem for Schur complements can be applied to matrices S of the speciic form (9.6) to conclude that the number of negative eigenvalues of the 2n 2n matrix S equals the sum of the numbers of negative eigenvalues of the two n n blocks ^ L _q _q and ^ L qq. For Lagrangians of the form (9.10) L(q; _ q) = 1 2 _ q T T (q) _ q + Q(q) T _ q V (q) ; that are quadratic in the velocities _ q with T > 0, we may further conclude that at equilibria, ^ L _q _q = ^ T has no negative eigenvalue, and ^ L qq = V qq (^ q). Thus for Lagrangian systems of the form (9.10), Theorems 1 and 2 can be applied with the number of negative eigenvalues of S being replaced with the number of negative eigenvalues of the Hessian ^ V qq of the potential. (Moreover Lemma 2 implies that hypothesis (2.2) is automatically satissed.) In particular it may be concluded that in the presence of complete Rayleigh dissipation, the only (nondegenerate) equilibria that are stable are minima of the potential, a result sometimes known as the Kelvin-Tait-Chetayev Theorem; see 10], Chapter 5.10. Rayleigh dissipation is also discussed in 4]. While the primary focus is on the case of relative equilibria, they do consider the case of equilibria. Their attention is restricted to the important, but nevertheless special, case of Hamiltonians of the form (9.9). They derive perturbation formulas, but only for the case of simple eigenvalues. They also adopt a Lyapunov-type approach to prove that at critical points that are not …